The theory of isogeny estimates for Abelian varieties
provides `additive bounds' of the form `

*d* is at most

*B*' for the
degrees

*d* of certain isogenies. We investigate whether these can
be improved to `multiplicative bounds' of the form `

*d* divides

*B*'.
We find that in general the answer is no (Theorem 1), but that
sometimes the answer is yes (Theorem 2). Further we apply the
affirmative result to the study of exceptional primes

in
connexion with modular Galois representations coming from
elliptic curves: we prove that the additive bounds for

of
Masser and Wüstholz (1993) can be improved to multiplicative
bounds (Theorem 3).